Low-frequency vibration isolation axial functional gradient porous beam structure and analysis method

Abstract

The invention relates to the technical field of mechanical structure vibration reduction and noise reduction, in particular to a low-frequency vibration isolation axial functional gradient porous beam structure and an analysis method, the structure is composed of a uniform beam area and a gradient porous beam area, the porosity in the uniform beam area section is 0, and the porosity of the gradient porous beam area section is distributed along the axial power law. The structure that the porosity changes in the axial direction in a gradient mode is adopted, the gradient change of the material characteristics (density and Young modulus) of the beam in the axial direction is achieved through the pores with the gradient change, and the energy gathering effect can be achieved through the characteristic that the impedance changes in the axial direction in a gradient mode. As vibration bending waves spread in a porous area, vibration energy is further dissipated and blocked, so that the efficient vibration reduction and isolation effect is achieved, and the situation that the weight and cost of the structure are increased due to the fact that a traditional acoustic black hole beam structure needs to be additionally laid with a viscoelastic damping material for energy absorption is avoided; and the problems of safety and use limitation caused by a gradually thinned appearance structure and the like are solved.

Description

Technical Field

[0001] This invention relates to the field of vibration reduction and noise reduction technology for mechanical structures, specifically to an axially functionally graded porous beam structure for low-frequency vibration isolation and its analysis method. Background Technology

[0002] With the rapid development of science and technology, higher requirements have been placed on vibration and noise control in engineering fields such as aerospace, mechanical engineering, and transportation. Traditional passive vibration isolation methods have extremely low vibration isolation performance in the low-frequency range, require a long stabilization time, have poor operability, and suffer from resonance problems. Active vibration isolation technology can significantly improve vibration isolation performance in the low-frequency range and avoid resonance. However, the control system is complex, requires external energy injection, and is relatively expensive.

[0003] Therefore, in recent years, an acoustic black hole structure has been proposed. This structure is achieved by gradually decreasing the thickness of a plate or beam according to a power-law function, causing a gradual change in structural impedance. This results in a decrease in the wave velocity and wavelength of bending waves within the structure, while the wave amplitude gradually increases. Ultimately, wave aggregation occurs at the apex of the acoustic black hole, effectively blocking vibration transmission. Due to the power-law change in the structure's thickness (… The inherent limitations of acoustic black hole structures in their manufacturing process create significant constraints. A certain degree of truncation inevitably occurs, leading to strong wave reflections at the truncation edges. While damping layers can be applied to the truncation edges to reduce reflection and concentrate energy, the variable thickness of the structure presents limitations in terms of safety and sealing. Summary of the Invention

[0004] The purpose of this invention is to provide an axially functionally graded porous beam structure for low-frequency vibration isolation and an analysis method thereof, aiming to solve the technical problems of truncation effect and manufacturing difficulties in acoustic black hole structure beams, and to improve safety and sealing performance.

[0005] To achieve the above objectives, this invention provides a low-frequency vibration isolation axially functionally graded porous beam structure, comprising a uniform beam region and a graded porous beam region. The porosity of the uniform beam region is 0, while the porosity of the graded porous beam region is distributed along an axial power law. The porosity of the axially functionally graded porous beam structure follows a piecewise function, expressed as follows:

[0006]

[0007] Where P represents the porosity function, with a value between 0 and 1; δ represents the porosity cutoff coefficient, which is related to the length of the gradient porous region and controls the maximum porosity value of the porous beam segment; N represents the porosity power law exponent, which determines the rate of change of the beam's porosity along the axial gradient; x is the axial length coordinate of the axial functional gradient porous beam structure; x0 to x1 is the length of the gradient porous beam segment; and x1 to x2 is the length of the uniform beam segment.

[0008] The Young's modulus and density of the axially functionally graded porous beam are functions of coordinate x and porosity P, as shown in the following formula:

[0009]

[0010] Wherein, E(x), Let E be the Young's modulus and density function of the gradient porous beam region. Uni , For the Young's modulus and density of a uniform beam region, when the porosity function P(x) approaches 0, what are the Young's modulus E(x) and density of a gradient porous beam region? The maximum value is reached at point x1. Similarly, when the porosity function P(x) approaches 1, the Young's modulus E(x) and density in the gradient porous beam region also increase. It reaches its minimum value at point x0.

[0011] The cumulative wave phase expression of the axially functionally graded porous beam structure is as follows:

[0012]

[0013] Where Φ is the cumulative wave phase, describing the integrated wave phase generated when a bending wave propagates from any point x on the beam to the end of the beam, i.e., the point of maximum porosity; h is the beam height; and L is the total length of the functionally graded porous beam. Let ω be the angular frequency, δ be the porosity cutoff coefficient, and N be the porosity power-law exponent. In the equation, when N ≥ 4, the equation diverges. This means that the cumulative wave phase approaches infinity; therefore, the wave never reaches its end and never reflects back, achieving an energy accumulation effect.

[0014] This invention also proposes an analytical method for a low-frequency vibration-isolated axially functionally graded porous beam structure, used to analyze and verify the aforementioned low-frequency vibration-isolated axially functionally graded porous beam structure, comprising the following steps:

[0015] Step 1: Analyze the traditional acoustic black hole beam wave focusing principle, derive the axial functionally graded porous beam wave number expression based on the Euler-Bernoulli beam wave number expression, further derive the bending wave cumulative wave phase integral expression, and propose the energy focusing condition of the axial functionally graded porous beam.

[0016] Step 2: Determine the porosity model based on the energy accumulation conditions, and derive the piecewise function expression of the material parameter characteristics based on the porosity model;

[0017] Step 3: Under the Rayleigh-Ritz framework, a semi-analytical dynamic model of an axially functionally graded porous beam is established using Gaussian functions as basis functions.

[0018] Step 4: The model of the uniaxial-biaxial functionally graded porous structure is verified from the perspectives of free vibration and forced vibration response.

[0019] Optionally, the material parameters in step 2 may include Young's modulus and density.

[0020] This invention provides an axially functionally graded porous beam structure for low-frequency vibration isolation and its analysis method. The structure consists of two parts: a uniform beam region and a graded porous beam region. The porosity of the uniform beam region is 0, while the porosity of the graded porous beam region is distributed along the axial direction using a power law. The structure of this invention features a porosity gradient along the axial direction. Utilizing the gradient porosity, the material properties (density and Young's modulus) of the beam can be varied axially, and its impedance can also vary axially, achieving an energy concentration effect. As the bending wave propagates in the porous region, the vibration energy is further dissipated and blocked due to the gradual change in the material's stiffness and damping characteristics, thus achieving a highly efficient vibration reduction and isolation effect. This avoids the problems of traditional acoustic black hole beam structures, which require additional viscoelastic damping materials for energy absorption, leading to increased structural weight and cost, and limitations in application. It also avoids the irregular nonlinear strain phenomena caused by manufacturing defects in traditional acoustic black hole structures. Attached Figure Description

[0021] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0022] Figure 1 This is a schematic diagram of the geometric model of an axially functionally graded porous beam structure for low-frequency vibration isolation according to an embodiment of the present invention.

[0023] Figure 2 This is a schematic diagram of a power-law pore model of an axially functionally graded porous beam structure for low-frequency vibration isolation according to an embodiment of the present invention.

[0024] Figure 3 This is a schematic diagram of the boundary constraints of a biaxial functionally graded porous beam according to an embodiment of the present invention.

[0025] Figure 4 This is a schematic diagram of the pore model distribution of a biaxial functionally graded porous beam according to an embodiment of the present invention.

[0026] Figure 5 This is a diagram showing the distribution of density and Young's modulus along the beam length of the pore model of the biaxial functionally graded porous beam according to an embodiment of the present invention.

[0027] Figure 6 This is a schematic diagram of the transfer rate of a biaxial functionally graded porous beam according to an embodiment of the present invention. Detailed Implementation

[0028] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0029] This invention provides an axially functionally graded porous beam structure for low-frequency vibration isolation, which consists of two parts: a uniform beam region and a graded porous beam region. The porosity of the uniform beam region is 0, while the porosity of the graded porous beam region is distributed along the axial power law.

[0030] Please see Figures 1 to 6 The following examples further illustrate this point:

[0031] In this invention, the porosity of the axially functionally graded porous beam follows a piecewise function, as shown in Equation 1:

[0032] (1)

[0033] In equation (1), P represents the porosity function, with a value between 0 and 1. δ represents the porosity cutoff coefficient, which is related to the length of the gradient porous region and controls the maximum porosity value of the porous beam segment. N represents the porosity power law exponent, which determines the rate of change of the beam's pores along the axial gradient.

[0034] x represents the axial length coordinate of the axially functionally graded porous beam structure, x0 to x1 represents the length of the graded porous beam region segment, and x1 to x2 represents the length of the uniform beam region segment.

[0035] Figure 2 This is a schematic diagram of a power-law porosity model of an axially functionally graded porous beam structure for low-frequency vibration isolation according to an embodiment of the present invention. The porosity of the uniform beam segment is 0, and the porosity of the porous beam segment varies along the axial direction in a power-law manner. The energy accumulation effect of the axially functionally graded porous beam is studied by varying the power-law exponent N of different porosities.

[0036] Equation (2) gives the distribution of Young's modulus and density of the beam. The beam's physical properties are: Young's modulus and density are functions of coordinate x and porosity P. By adjusting the magnitude of the exponent N, the axial material properties of the beam can be changed. Figure 3 .

[0037] Figure 3 This is a schematic diagram of the boundary constraints of a biaxial functional gradient porous beam according to an embodiment of the present invention. Displacement springs and rotation springs are used to simulate different boundary conditions of the beam. (1) When the stiffness values ​​of the displacement springs and rotation springs approach 0, (2) the two ends of the beam are free boundaries. When the stiffness values ​​approach infinity, the two ends of the beam are fixed boundaries. (3) When the stiffness of the displacement springs approaches infinity and the stiffness of the rotation springs approaches 0, the two ends are simply supported boundaries.

[0038] (2)

[0039] According to the energy accumulation condition of traditional acoustic black hole structures, the cumulative phase of the bending wave propagating from any point on the beam to the end of the beam can be expressed by equation (3):

[0040] (3)

[0041] Cumulative phase It reflects the total change in phase of a curved wave along its propagation path, and it is related to factors such as the propagation distance of the wave, the properties of the medium, and the frequency of the wave. The wave number represents the position-dependent wave number. In the structure of an acoustic black hole, when the thickness approaches zero, the bending wave velocity also approaches zero. The cumulative phase and amplitude of the wave will increase infinitely, causing the bending wave to concentrate at the tip of the structure, resulting in acoustic wave trapping and energy accumulation effects.

[0042] Based on the Euler-Bernoulli expression for the wave number of a beam, the expression for the wave number of an axially functionally graded porous beam is derived. Substituting this into equation (3) and rearranging, the cumulative wave phase expression for the axially functionally graded porous beam is derived as shown in equation (4). From this equation, it can be concluded that when N is greater than or equal to 4, the cumulative wave phase... The value will approach infinity, indicating that the bending wave in the beam structure cannot reach the boundary of the functionally graded porous beam structure, thereby eliminating wave reflection at the edge and achieving an energy concentration effect.

[0043] (4)

[0044] Furthermore, this invention also proposes an analytical method for a low-frequency vibration-isolated axially functionally graded porous beam structure, used to analyze and verify the aforementioned low-frequency vibration-isolated axially functionally graded porous beam structure, comprising the following steps:

[0045] Step 1: Analyze the traditional acoustic black hole beam wave focusing principle, derive the axial functionally graded porous beam wave number expression based on the Euler-Bernoulli beam wave number expression, further derive the bending wave cumulative wave phase integral expression, and propose the energy focusing condition of the axial functionally graded porous beam.

[0046] Step 2: Determine the porosity model based on the energy accumulation conditions, and derive the piecewise function expression of the material parameter characteristics based on the porosity model;

[0047] Step 3: Under the Rayleigh-Ritz framework, a semi-analytical dynamic model of an axially functionally graded porous beam is established using Gaussian functions as basis functions.

[0048] Step 4: The model of the uniaxial-biaxial functionally graded porous structure is verified from the perspectives of free vibration and forced vibration response.

[0049] The material parameters in step 2 include Young's modulus and density.

[0050] Based on the aforementioned conditions for energy accumulation in axially functionally graded porous beam structures, this paper will focus on axially and biaxially functionally graded porous beams, establishing their dynamic analysis models based on energy principles. From both free and forced vibration perspectives, the porosity exponent N and the cutoff coefficient will be revealed. The influence of axially functionally graded porous beams on energy accumulation and vibration transmission characteristics was investigated. This study verifies the application potential of axially functionally graded porous beams in low-frequency applications, vibration reduction, and vibration isolation.

[0051] The structural parameters provided in this embodiment are determined according to actual needs. Specifically, this embodiment may use the following dimensions: beam length * width * height is 240mm * 1mm * 3mm. Figure 4 As shown, Figure 4 This is a schematic diagram of the pore model distribution of a biaxial functionally graded porous beam according to an embodiment of the present invention. It is composed of... Figure 1 Axially functionally graded porous beams are symmetrically obtained, and the material properties (Young's modulus and density) are distributed in large and small sizes. Based on the goal of achieving energy concentration effect, the vibration reduction and isolation characteristics of biaxially functionally graded porous beams are studied.

[0052] The beam has uniform material at both ends and a symmetrical porous structure with pores distributed along the axial power law in the middle. The entire beam structure is made of steel with a density of 7800 kg / m³, Young's modulus of 210e9 Pa, Poisson's ratio of 0.3, and loss factor of 0.003. To evaluate the vibration isolation level of the biaxial functionally graded porous beam, point A is selected as the excitation source, the root mean square value of acceleration at point B is extracted, and its vibration transmissibility is calculated. The transmissibility TL is calculated using the following formula:

[0053] (5)

[0054] Where awB is the root mean square value of the transverse acceleration at response point B, and awA is the root mean square value of the acceleration at excitation point A. This is to reveal the porosity exponent N and the cutoff coefficient. The effect on the level of vibration energy transfer is determined by the cutoff factor. The value is 0.01, and the porosity index N is 2, 3, 4, and 5. Its vibration transmissibility is as follows: Figure 4 As shown in the figure, when TL is less than 0, it indicates that the vibration energy is dissipated through the gradient porous region, and the vibration level on the response side decreases. The figure shows that when the porosity N is greater than or equal to 4, the vibration transmissibility of the biaxial functionally graded porous beam within the range of 0~500Hz is less than 50dB, demonstrating its superior vibration isolation level. This is because the biaxial functionally graded porous beam, based on its gradient material properties, can concentrate vibration energy at the middle position of the beam, blocking the forward transmission of vibration energy while reducing energy reflection.

[0055] Figure 5 This is a diagram showing the distribution of density and Young's modulus along the beam length of the pore model of the biaxial functionally graded porous beam according to an embodiment of the present invention.

[0056] Figure 6 This is a schematic diagram of the transmissibility of a biaxial functionally graded porous beam according to an embodiment of the present invention. The figure shows the transmissibility curves TL for different porosity power-law exponents N of 2, 3, 4, and 5. It can be clearly seen that when the porosity exponent N is greater than or equal to 4, it has a significant effect on blocking vibration transmission in the 0-500Hz range. In particular, when N is 5, the transmissibility TL value is close to -80dB, demonstrating the vibration reduction and isolation capability of the biaxial functionally graded porous beam in the low-frequency range.

[0057] Furthermore, the present invention can employ rapid prototyping 3D printing, and the lightweighting of the structure can be further achieved through topology optimization of the porous structure.

[0058] In summary, this invention provides an axially functionally graded porous beam structure with energy concentration effect. It combines the energy concentration characteristics of traditional acoustic black hole structures with the advantages of lightweight, easy manufacturing, and rapid prototyping using 3D printing technology, avoiding the limitations of traditional acoustic black hole beam structure manufacturing processes. Furthermore, this invention reduces the limitations of traditional acoustic black hole structures in vibration control and noise reduction applications. Topology optimization of the porous structure can achieve lightweighting and improve the overall mechanical performance of the structure.

[0059] The above description discloses only one or more preferred embodiments of the present invention, and should not be construed as limiting the scope of the present invention. Those skilled in the art will understand that all or part of the processes of the above embodiments can be implemented, and equivalent changes made in accordance with the claims of the present invention are still within the scope of the invention.

Claims

1. A low-frequency vibration-isolated axially functionally graded porous beam structure, characterized in that, It consists of two parts: a uniform beam region and a gradient porous beam region. The porosity of the uniform beam region is 0, while the porosity of the gradient porous beam region is distributed along the axial direction using a power law. The porosity of the axially functionally gradient porous beam structure follows a piecewise function, expressed as follows: ； Where P represents the porosity function, with a value between 0 and 1; δ represents the porosity cutoff coefficient, which is related to the length of the gradient porous region and controls the maximum porosity value of the porous beam segment; N represents the porosity power law exponent, which determines the rate of change of the beam's porosity along the axial gradient; x is the axial length coordinate of the axial functional gradient porous beam structure; x0 to x1 is the length of the gradient porous beam segment; and x1 to x2 is the length of the uniform beam segment.

2. The low-frequency vibration isolation axially functionally graded porous beam structure as described in claim 1, characterized in that, The Young's modulus and density of the axially functionally graded porous beam are functions of coordinate x and porosity P, as shown in the following formula: ； Wherein, E(x), Let E be the Young's modulus and density function of the gradient porous beam region. Uni , For the Young's modulus and density of a uniform beam region, when the porosity function P(x) approaches 0, what are the Young's modulus E(x) and density of a gradient porous beam region? The maximum value is reached at point x1. Similarly, when the porosity function P(x) approaches 1, the Young's modulus E(x) and density in the gradient porous beam region also increase. It reaches its minimum value at point x0.

3. The low-frequency vibration isolation axially functionally graded porous beam structure as described in claim 2, characterized in that, The cumulative wave phase expression of the axially functionally graded porous beam structure is as follows: ； Where Φ is the cumulative wave phase, describing the integrated wave phase generated when a bending wave propagates from any point x on the beam to the end of the beam, i.e., the point of maximum porosity; h is the beam height; and L is the total length of the functionally graded porous beam. δ represents the porosity cutoff coefficient, and N represents the porosity power law exponent; in the formula, when N≥4, the equation diverges.

4. An analytical method for a low-frequency vibration-isolated axially functionally graded porous beam structure, used to analyze and verify the low-frequency vibration-isolated axially functionally graded porous beam structure as described in any one of claims 1 to 3, characterized in that, Includes the following steps: Step 1: Analyze the traditional acoustic black hole beam wave focusing principle, derive the axial functionally graded porous beam wave number expression based on the Euler-Bernoulli beam wave number expression, further derive the bending wave cumulative wave phase integral expression, and propose the energy focusing condition of the axial functionally graded porous beam. Step 2: Determine the porosity model based on the energy accumulation conditions, and derive the piecewise function expression of the material parameter characteristics based on the porosity model; Step 3: Under the Rayleigh-Ritz framework, a semi-analytical dynamic model of an axially functionally graded porous beam is established using Gaussian functions as basis functions. Step 4: The model of the uniaxial-biaxial functionally graded porous structure is verified from the perspectives of free vibration and forced vibration response.

5. The analytical method for low-frequency vibration isolation axially functionally graded porous beam structures as described in claim 4, characterized in that, The material parameters in step 2 include Young's modulus and density.

Images